Remark 8.5 Under the assumptions of Theorem REF , by the scattering theory (see e.g. Hörmander [1]}), the spectrum \(\sigma (H)\)  consists of finitely many negative eigenvalues \(\lbrace \lambda _k\rbrace _{k=1}^{N}\)  with finite multiplicity and the absolutely continuous spectrum \(\sigma _{ac}(H)=[0, \infty )\) . In particular, \(H\)  does not have neither embedded positive eigenvalues nor singular spectrum. Hence by the spectral theorem and the intertwining property \([{intertwining_1}]{\textup {{\!\!\ref *{intertwining_1}}}}\) , we can write down
\(f(H)=\sum _{j=1}^Nf(\lambda _j)P_{\lambda _j}+W_\pm f(\Delta ^2)W_\pm ^*,\) 
